Concise - Self test 5

Question 1

Why was an elliptic planform for a wing considered to be advantageous?

Answer 1

• Elliptic spanwise lift distribution yields minimum induced drag
• Requires elliptic spanwise chord variation without geometric/aerodynamic twist
• Achieves minimum induced drag

Question 2

What is geometric twist in the context of a finite wing and explain the benefit and challenge it represents.

Answer 2

• Geometric twist: angle of attack/chordline varies with spanwise position
• Purpose: achieve spanwise lift distribution to minimize induced drag (e.g., elliptic circulation)
• Challenge: constructing twisted structures is difficult with metal airframes

Question 3

Explain why birds flying in a V-formation is advantageous.

Answer 3

• V-formation allows trailing tip vortices from adjacent birds to overlap
• Counter-rotating vortices cancel each other
• Reduces downwash effect, leading to less induced drag and lower power requirements
• Leading bird experiences downwash; trailing positions experience upwash
• Not all positions in the V formation receive equal benefits

Question 4

State Helmholz’s third theorem that governs vortex filaments and explain the consequence for flow around a wing of finite span.

Answer 4

• Helmholtz’s Third Theorem:
  • In absence of external rotational forces, initially irrotational fluid remains irrotational
  • Circulation around fluid particles initially zero remains zero
• Consequence for finite wings:
  • To generate lift (non-zero circulation), an opposite circulation (starting vortex) must be shed from the trailing edge
  • Starting vortex convects downstream and its effect diminishes over time, negligible in steady conditions

Question 5

What are the limitations of the Prandtl lifting line theory for 3D wings (i.e. stacked horse vortices)?

Answer 5

• Does not account for viscous flow or swept wings
• Assumes shed vorticity sheet remains planar
• Limitations:
  • Vorticity sheet actually rolls into single wing tip vortices
  • Wing tip vortices induce downward velocity in each other, causing trailing vortices to be below the wing
  • Trailing vortices are not in the same plane

Question 6

How are Helmholtz’ theorems for the motion of vortex filaments satisfied for a 2D aerfoil?

Answer 6

• 2D vortex filament slice is a point vortex
• Vortex filaments are parallel to the z-axis in 2D flow
• Helmholtz’s First and Second Theorems:
  • Constant strength filaments
  • Filaments extend to +/- infinity
• Third Theorem:
  • No removal or change in vortex strength over time
  • Satisfied in inviscid flow

Question 7

State Helmhotz’s first two theorems governing vortex filaments and explain the consequence for flow around a wing of finite span.

Answer 7

• Helmholtz’s First Two Theorems:
  
  1. Vortex filament strength is constant along its entire length
  2. Vortex filament cannot end in fluid; must extend to boundaries or form closed loops
• Consequence:
  • Lifting wings generate wing tip vortices extending downstream towards infinity

Question 8

What is aerodynamic twist in the context of a finite span wing and explain the purpose of this feature.

Answer 8

• Aerodynamic twist: shape (camber) of aerofoil varies with spanwise position
• Purpose:
  • Achieve spanwise lift distribution to minimize induced drag (e.g., elliptic circulation)
  • Help avoid stall on specific parts of the wing